5.4 Indefinite Integrals and the Net Change Theorem/3

${\begin{aligned}&\int \cos ^{3}xdx=\sin {x}-{\frac {1}{3}}\sin ^{3}x+C\\[2ex]&{\frac {d}{dx}}{[\sin {x}-{\frac {1}{3}}\sin ^{3}{x}+C]}\\[2ex]&={\cos {x}-{\frac {1}{3}}\cdot 3\sin ^{2}{x}\cos {x}+0}\\[2ex]&=\cos {x}-\sin ^{2}{x}\cos {x}\\[2ex]&=\cos {x}-(1-cos^{2}(x))\cos {x}\\[2ex]&=\cos ^{3}{x}\end{aligned}}$

Note: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1-\cos^2(x) = \sin^2(x) }